Scaling and fractals in hydrology

Virtually, all areas of hydrology have been deeply influenced by the concepts of fractality and scale invariance. The roots of scale invariance in hydrology can be traced to the pioneering work of Horton, Shreve, Hack and Hurst on the topology and metric properties of river networks and on river flow. This early work uncovered symmetries and laws that only later were recognized as manifestations of scale invariance. Le Cam, who in the early 1960s pioneered the development of multi-scale pulse models of rainfall, provided renewed impetus to the use of scale-based models. Fractal approaches in hydrology have become more rigorous and widespread since Mandelbrot systematized fractal geometry and multifractal processes were discovered. This chapter reviews the main concepts of fractality and scale invariance, the construction of scale-invariant processes, their properties, and the inference of scale invariance from data. We highlight the recent developments in four areas of hydrology: rainfall, fluvial erosion topography, river floods, and flow through porous media.